Rosenau-Burgers方程的修正局部Crank-Nicolson格式（英文）

本文给出了Rosenau-Burgers方程的两种修正局部Crank-Nicolson格式.首先,求解原有的偏微分方程对空间方向进行有限差分离散而得到的常微分方程.其次,利用矩阵分裂技术对这个方程的指数系数矩阵分别按行和元素进行逼近.最后,利用修正局部Crank-Nicolson方法得到了两种格式.讨论了格式的稳定性、收敛性和先验误差估计.数值实验结果表明了理论证明的正确性及格式的有效性.该格式具有结构简单、精度高的优点.     

Rosenau-Burgers 方程的修正局部 Crank-Nicolson 格式

本文给出了 Rosenau-Burgers 方程的两种修正局部 Crank-Nicolson 格式．首先,求解原有的偏微分方程对空间方向进行有限差分离散而得到的常微分方程．其次,利用矩阵分裂技术对这个方程的指数系数矩阵分别按行和元素进行逼近．最后,利用修正局部 Crank-Nicolson 方法得到了两种格式．讨论了格式的稳定性、收敛性和先验误差估计．数值实验结果表明了理论证明的正确性及格式的有效性．该格式具有结构简单、精度高的优点．     

时间分数阶扩散方程的一种交替分带并行差分方法

分数阶反常扩散方程具有深刻的物理背景和丰富的理论内涵,其数值解法的研究具有重要的科学意义和工程应用价值．针对二维时间分数阶反常扩散方程,本文研究一种交替分带 Crank-Nicolson 差分的并行计算方法（ABdC-N 方法）．该格式是在交替分带技术的基础上,结合经典显式、隐式和 Crank-Nicolson 差分格式构造而成．理论分析和数值试验表明,ABdC-N 方法是无条件稳定和收敛的,具有良好的计算精度和并行计算性质,并且计算效率远优于经典的串行差分方法,证实本文 ABdC-N 差分方法求解二维时间分数阶反常扩散方程是有效的．     

对流扩散方程的特征线EFG算法

为了消除对流扩散方程因对流占优引起的数值震荡,本文首先将其转化为特征形式,并利用移动最小二乘基函数,构建了特征线无单元Galerkin方法．再对新建方法进行收敛性分析,分别给出关于支持域半径和时间步长的两种误差估计．最后,分别针对一维和二维算例进行了数值计算,并与有限元法进行了比较．数值结果表明,本文算法收敛性好,可以消除数值震荡,且通过选取合适的罚因子和支持域的无量纲尺寸,计算精度比有限元法更高,是求解对流占优扩散方程的一种有效程数值计算方法．     

求解变系数对流扩散反应方程的指数型高精度紧致差分方法

本文给出了一种数值求解变系数对流扩散反应方程的指数型高精度紧致差分方法.我们首先将模型方程变形,借助常系数对流扩散方程的指数型高精度紧致差分格式,采用残量修正法得到变系数对流扩散反应方程的指数型高精度紧致差分格式;并从理论上分析了当Pelect数很大时,本文格式达到四阶计算精度时网格步长的限制条件;离散得到的代数方程组可采用追赶法直接求解.数值实验结果与理论分析完全吻合,表明了本文格式对于边界层问题或大梯度变化的物理量求解问题具有的高精度和鲁棒性的优点.     

体积约束的非局部扩散问题基于新的技巧的有限元方法

体积约束的非局部扩散问题在复合材料的断裂、多晶体的断裂、纳米纤维网络、裂缝的不稳定、图像处理等领域有重要应用,现存的数值方法精度不高。因此,设计一种高阶的有限元方法来求解二维体积约束的非局部扩散问题是十分必要的,但需克服维数增加带来的自由度骤增的困难。为此,采用了一种新技巧计算线性元的刚度矩阵,该数值方法的刚度矩阵是从一个新的矩阵$B$中提取的,该矩阵易于计算,并给出了单元的编码原理和数值计算节点的编码表达式,并通过数值算例验证了该方法对二维体积约束的非局部扩散问题具有几乎最优收敛阶。值得一提的是,求解二维体积约束的非局部扩散问题并不是平凡的。     

一维非定常对流扩散方程非均匀网格上的高精度紧致差分格式

本文在非均匀网格上给出了求解非定常对流扩散方程的一种高精度紧致差分格式,特别适合边界层和大梯度等问题的求解.从稳态对流扩散方程入手,首先,基于非均匀网格上的泰勒级数展开对空间导数项进行离散,然后对时间项采用二阶向后欧拉差分公式,从而得到一维非定常对流扩散方程在非均匀网格上的三层全隐式紧致差分格式.新格式在时间具有二阶精度,空间具有三到四阶精度,并且是无条件稳定的.最后,通过数值实验验证了本文格式的精确性,以及在处理诸如边界层和大梯度问题上的优势.     

基于多重有限体积法解决含有耗散项的双曲扩散问题

双曲扩散方程在数学上是一类重要的偏微分方程,在众多工程领域中有着广泛的应用.其常用于描述声波和电势场中电流的传播问题,也用于模拟计算流体力学中的对流扩散以及热传导模型.本文研究了一类具有耗散项的双曲扩散方程,并通过数值方法对其初边值问题进行求解.针对已有的有限体积法在求解该问题时精度无法提高的问题,本文基于变限积分的方法提出了新的高精度有限体积格式,并利用Fourier分析法得到其离散格式无条件稳定的结论,最后利用数值实验证实了理论分析结果.     

二维 Volterra-Fredholm 积分方程数值算法研究及收敛性分析

针对一类二维非线性Volterra-Fredholm积分方程,提出利用二维Block-Pulse函数为基函数进行数值求解。首先,引入Block-Pulse函数的定义及基函数的向量表示形式;其次,根据二维Block-Pulse函数的不相交性和正交性推导了基向量的积分算子矩阵和乘积算子矩阵;然后,基于该算子矩阵将待求问题转化为一系列向量的乘积形式,利用配点法离散未知变量获得原问题的数值解;最后,通过两个具体的数值算例对所提算法的可行性和收敛性进行了验证。     

色噪声激励下Duffing—Rayleigh—Mathieu系统的稳态响应

基于广义谐和函数与随机平均原理,研究了具有强非线性的Duffing-Rayleigh-Mathieu系统在色噪声激励下的稳态响应.通过van der Pol坐标变换,将系统运动方程转化为关于幅值与初始相位角的随机微分方程.应用Stratonovich-Khasminskii极限定理,作随机平均,得到近似的二维扩散过程.在此基础上,考虑共振情形,引入相位差变量,做确定性的平均,得到关于幅值与相位差的It(o)随机微分方程.建立对应的Fokker-Planck-Kolmogorov(FPK)方程,结合边界条件与归一化条件,用Crank-Nicolson型有限差分法求解稳态的FPK方程,得到平稳状态下系统的联合概率分布.用Monte Carlo数值模拟法验证了理论方法的有效性.     

Recursive solution of inverse natural convection problems by means of mode reduction

 A recursive method based on the Kalman filtering is developed to solve inverse natural convection problems of estimating the unsteady nonuniform wall heat flux from temperature measurements in the flow. By employing the Karhunen–Loève Galerkin procedure that reduces the Boussinesq equation to a small set of ordinary differential equations, the computational difficulties associated with the Kalman filtering for the partial differential equations are overcome. The present method is assessed through several numerical experiments, and is found to yield satisfactory results. Received 20 January 2001 / Accepted 31 May 2001     

Theoretical analysis on mixed convection boundary layer flow over a wedge with uniform suction or injection

Summary Forced and free mixed convection boundary layer flow over a wedge with uniform suction or injection is theoretically investigated. Nonsimilar partial differential equations are transformed into ordinary differential equations by means of difference-differential method. The solutions of the resulting equations are obtained in integral forms and are calculated by iterative numerical procedures. The results were given for velocity profiles, temperature profiles, friction and heat transfer parameters for various values of suction/injection parameter, pressure gradient parameter and buoyancy parameter.     

An Accelerated Two-Dimensional Unsteady Heat Conduction Calculation Procedure for Thermal-Conductivity Measurement by the Transient Short-Hot-Wire Method

A fast and accurate procedure is proposed for solution of the two-dimensional unsteady heat conduction equation used in the transient short-hot-wire method for measuring thermal conductivity. Finite Fourier transforms are applied analytically in the wire-axis direction to produce a set of one-dimensional ordinary differential equations. After discretization by the finite-volume method in the radial direction, each one-dimensional algebraic equation is solved directly using the tri-diagonal matrix algorithm prior to application of the inverse Fourier transform. The numerical procedure is shown to be very accurate through comparison with an analytical solution, and it is found to be an order of magnitude faster than the usual numerical solution.     

Forced and free mixed convection boundary layer flow with uniform suction or injection on a vertical flat plate

Summary A laminar forced and free mixed convection flow on a flat plate with uniform suction or injection was theoretically investigated. Nonsimilar partial differential equations are transformed into nonsimilar ordinary ones by means of difference-differential method. The solutions of the resulting equations are obtained with integral forms, and are calculated by the method of successive iteration. The velocity profiles, temperature profiles, friction coefficient and heat transfer coefficient are obtained for various values of suction/injection parameter and buoyancy parameter.     

Thermal boundary layers in magnetohydrodynamic flow over a flat plate in the presence of a transverse magnetic field

Summary A boundary layer solution for the heat transfer of an electrically conducting fluid over a semi-infinite flat plate in the presence of a transverse magnetic field has been studied. The heat due to viscous dissipation and stress work were also included into the energy equation. The governing nonsimilar partial differential equations are transformed into ordinary differential ones by means of difference-differential method. The temperature profiles and heat transfer coefficient are obtained for various values of the parameters entering the problem.     

Similarity solutions of unsteady boundary layer equations of a special third grade fluid

Two-dimensional, unsteady, laminar boundary layer equations of a special model of non-Newtonian fluids are considered. The fluid can be considered as a special type of power-law fluid. The problem investigated is the flow over a moving surface, with suction of injection. Two different type of ordinary differential equations system are found using the transformations. Using scaling and translation transformations, equations and boundary conditions are transformed into a partial differential system with two variables. Using translation and a more general transformation, the boundary value problem is transformed into an ordinary differential equations system. Finally, we numerically solve two different ordinary differential equations, separately.     

热弹性动力学耦合问题的插值型移动最小二乘无网格法研究

该文基于插值型移动最小二乘法,将无网格局部Petrov-Galerkin（MLPG）法用于二维耦合热弹性动力学问题的求解。修正的Fourier热传导方程和弹性动力控制方程通过加权余量法来离散,Heaviside分段函数作为局部弱形式的权函数,从而得到描述热耦合问题的二阶常微分方程组。然后利用微分代数方法,温度和位移作为辅助变量,将上述二阶常微分方程组转换成常微分代数系统,采用Newmark逐步积分法进行求解。该方法无需Laplace变换可直接得到温度场和位移场数值结果,同时插值型移动最小二乘法构造的形函数由于满足Kroneckerdelta特性,因此能直接施加本质边界条件。最后通过两个数值算例来验证该方法的有效性。     

Characteristic finite analytic method (CFAM) for incompressible Navier-Stokes equations

Summary Along its characteristics, the Navier-Stokes equations are reduced to a diffusion equation. In this work, the analytic solution of the diffusion equation is applied to construct its finite analytic method (FAM) in space-time domain. Then, it is applied along those characteristics to efficiently simulate the Navier-Stokes equations. CFAM is therefore introduced. This method is verified by 2-D and 3-D driven cavity flows. A comparison of our result with existing computational data is made, and good agreement is found. The result also shows many details of 2-D and 3-D streamline structures.     

不等温壁竖平板层流自然对流换热问题探讨

本文采用摄动法求解不等温竖平板层流自然对流的换热问题。通过正则摄动将很难处理的偏微分方程组转化为常微分方程组,并对边界条件进行相应变换。采用四阶龙格——库塔方法进行数值计算。计算中采用 Nachtseim——Swigert及 Marquardt 方法进行选代以确保渐近边界条件的满足及其数值计算的收敛性。由此得到竖平板自然对流为层流流动时在任意壁面处的近似换热量,可把它应用于相应的工程换热问题的计算中。     

On the isovectors of a class of nonlinear diffusion equations

A class of nonlinear diffusion equations is expressed in terms of an ideal of exterior differential forms. The components of the associated isovector field are constructed by using its transport property under Lie's derivative. The solution of the corresponding orbital equations generate an invariant group of transformations which reduce the nonlinear diffusion equation to an ordinary differential equation. For the problem of heat propagation, in an initially cool infinite media, due to a plane source, an exact solution is derived.